Devyani Khobradage, the Indian official whose arrest led to new tension between the US and India, has left New York after being indicted on charges of visa fraud pertaining to wages for her housekeeper. There are some circumstances under which she could yet be prosecuted.

Devyani Khobradage, a consular officer for India, was arrested last month in New York on charges of fraudulently filing a US visa application so her housekeeper could come to America to work. She left the US Thursday after the federal government issued a criminal indictment against her. Ms. Khobradage is shown here at a December fundraising event on Long Island four days before her arrest.

Mohammed Jaffe/AP

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The Indian diplomat at the center of a diplomatic feud between the US and India boarded a plane back to New Delhi Thursday evening, after the US formally indicted her on charges of visa fraud, Indian officials said on Friday. Her departure, arranged through an agreement between Indian and US officials, appeared to be an effort by both countries to settle a spat that has rankled bilateral relations, though each continues to maintain that it is in the right.

Devyani Khobradage, a consular officer, was arrested Dec. 12 in New York on charges of filing a US visa application for her housekeeper, Sangeet Richard, that falsely claimed intentions to pay the woman wages in accordance with US labor laws. Ms. Khobradage's arrest, in which she was strip-searched and held in a cell with what the Indian press called drug addicts, sparked ire in India that unfurled into massive street protests and reverberated in retaliatory measures by India against US diplomats.

The incident soon evolved into a battle for the moral high ground, as the US pulled away from Secretary of State John Kerry’s initial expressions of “regret” and indicated that it would not drop its allegations against India’s diplomat. On Thursday, US Attorney Preet Bharara, the federal prosecutor for the Southern District of New York, announced that Khobradage would be indicted on charges of visa fraud and making false statements – a sign that efforts to arrange a plea bargain had come to nothing.

It is uncertain whether Khobradage will ever be prosecuted. On Wednesday, the US State Department heeded India’s request to grant Khobradage full diplomatic immunities – a status that would shield her from prosecution and required upgrading her rank. The US then asked India to waive her immunity so the indictment could be pursued, but India declined. At that point, the US asked that Khobradage return to India. The Indian Ministry of Foreign Affairs has said it will transfer Khobradage to a position at its New Delhi office.

Khobradage can now be prosecuted only if she returns to the US without the diplomatic immunity the US has granted her. Her American husband and children still live in the US, suggesting that the battle might not be over.

The US decision to give Khobradage diplomatic privileges, and India’s decision to pull her from the country, seems to be a tacit effort to put a stop, at least temporarily, to the deterioration of US-India relations during the past, bitter month. The effect is a stalemate in which neither country admits wrongdoing, but in which Khobradage does not face immediate prosecution.

Khobragade is accused of submitting an employment form to the US Embassy in India that listed wages for her housekeeper, Ms. Richards, at $9.75 per hour – a sum that complies with New York State labor laws. But the agreement Khobragade gave to her housekeeper, and did not submit to US authorities, offered wages of just $6,876 per year, or about $1.42 per hour, given that Richards worked between 94 and 104 hours a week. The Embassy processed a visa for Richards on the basis of that falsified employment form, according to the indictment.

In June 2013, after Khobradage allegedly declined Richards' repeated requests to end her employment and to have her passport returned, Richards fled her employer's home for Safe Horizon, a nonprofit group that represents victims of human trafficking. One of Khobradage’s relatives is alleged to have made repeated calls to Richards’ husband in India, in hopes that the calls would “intimidate” Richards into returning to India without going to the US authorities with her story, the indictment alleges. Khobradage was subsequently arrested in December.

On Thursday, Indian officials continued to assert that the diplomat had done nothing wrong and insisted that the US had committed the more egregious act by detaining her.

“The worst that can be said about her is that she did not comply with the amounts that was supposed to be paid under your [US] law,” Foreign Minister Salman Khurshid of India told The New York Times. “I don’t think that justifies treating her like a common criminal.”

He asked that the charges against Khobragade be dropped, alleging that there was no “legitimate legal ground for pursuing this case,” echoing national sentiment that Khobradge’s alleged crime is a negligible one, at least under Indian custom.

But US prosecutors also held firm to their position. In a letter to US District Judge Shira Scheindlin on Thursday, Mr. Bharara wrote that “the charges will remain pending” until Khobradage returns to the US without diplomatic privileges.

“We will alert the court promptly if we learn that the defendant returns to the United States in a nonimmune capacity, at which time the government will proceed to prosecute this case and prove the charges in the indictment,” the federal prosecutor wrote.

In a statement released by Safe Horizons on Friday, Richards expressed gratitude for the indictment.

“I would like to tell other domestic workers who are suffering as I did – you have rights and do not let anyone exploit you,” she said.